stacksortable

stacksortable is a conceptual term that describes the ability of data structures to be sorted efficiently using a stack as an auxiliary data structure. This means that an algorithm can take an unsorted sequence, process it using a stack (pushing and popping elements), and end up with a sorted sequence.

The primary requirement for a sequence to be stacksortable is that it must not contain a specific forbidden pattern. For arrays, this forbidden pattern is often described as a subsequence that is "231-avoiding" or "312-avoiding" in its relative ordering of elements. If an array contains a subsequence of three elements $a_i, a_j, a_k$ such that $i < j < k$ and $a_k < a_i < a_j$, then the array is not stacksortable. Similarly, if $i < j < k$ and $a_j < a_k < a_i$, it is also not stacksortable.

Algorithms that can perform stacksort typically involve iterating through the input sequence and making decisions about whether to push an element onto the stack or output it directly. The goal is to maintain a certain order on the stack that will eventually allow for the sorted output. The number of distinct permutations that are stacksortable is related to Catalan numbers.

The concept of stacksortability is relevant in theoretical computer science, particularly in the study of permutations and their relationships with stack operations and sorting networks. It helps in understanding the limitations and capabilities of stack-based sorting mechanisms.